Note on the Electron EDM

Transcription

1 Note on the Electron EDM W R Johnson October 25, 2002 Abstract Ths s a note on the setup of an electron EDM calculaton and Schff s Theorem 1 Basc Relatons The well-known relatvstc nteracton of the electron s anomolous magnetc moment δµ wth a magnetc feld B s gven by H I = δµ β Σ B If we now assume that the electron has an electrc dpole moment EDM) d, the correspondng relatvstc nteracton wth an electrc feld E s gven by H I = d βσ E Ths term s obvously rotatonaly nvarant and therefore conserves angular momentum; however, t volates both party and tme-reversal symmetry Σ s even and E s odd under party; whereas, Σ s odd and E s even under tme reversal In an external feld E ext, the many-body Hamltonan for electrons charge e and EDM d) s, H = [h 0 )+V nuc )] e 2 r d β Σ E nt [e r + dβ Σ ] E ext 1) where wth V = ee nt = V, V nuc )+ 1 2 s the electrc feld at the th electron The terms on the frst lne of Eq 1) gve the modfed many-electron Hamltonan n the absence of external felds, e 2 r 1

2 whle those on the second lne descrbe the nteracton of the atomc electrons wth an external electrc feld For an external electrc feld of strength E n the z drecton, the atom-feld nteracton energy of an atom n a state v wth proecton m v s [ ] W = vm v ez + dβ Σ 3 vm v E = m v DE v Ths equatom serves to defne the atomc dpole moment D Thus [ ] D = v v ez + dβ Σ 3 v v 2) The frst term n ths expresson vanshes when evaluated wth wave functons of the unperturbed Hamltonan because of party conservaton If we expand the wave functon perturbatvely, keepng terms of order d, we fnd that D = D 0) + D 1) wth D 0) = d v v β Σ 3 v v > 3) D 1) = d n + d n v v z n n β Σ V v v v v β Σ V n n z v v 4) In the second term, we replace p to fnd v v ] z n n [ β Σ p,v v v D 1) = d n + d n ] v v [ β Σ p,v n n z v v In ths expresson, we may replace V = H H 0 where H 0 = [ c α p + βmc 2] 2

3 The part of D 1) from H may be evaluated usng completeness: D a 1) = d [ v v z n n β Σ p v v n v v β Σ p n n ] z v v [ = d v v z, β Σ p ] v v = d v v β Σ 3 v v = D 0) From ths, t follows that the sum D 0) +D a 1) = 0 Therefore, the atomc dpole moment reduces to D d v v [ ] z n n β Σ p,h 0 v v E n v E n + d n [ ] v v β Σ p,h 0 n n z v v 5) One can smplfy ths expresson usng ] [ [βσ p,h 0 βσ p, cα p + mc 2 β )] = [ ) ) ) )] σ p 0 0 σ p 0 σ p σ p 0 c 0 σ p σ p 0 σ p 0 0 σ p [ ) ) ) )] + mc 2 σ p σ p 0 0 σ p σ p ) =2cp =2cp 2 βγ ) Here, γ 5 = We are thereby led to ntroduce the effectve one-partcle) edm Hamltonan ) H edm = 2d c p 2 β γ 5 ) = H edm The expresson for the atomc dpole moment then takes the form D = n v v Z n n H edm v v + n v v H edm n n Z v v, 7) 3

4 where Z = z Snce H edm s Hermetan, we may wrte D =2 n v v Z n n H edm v v It s worth mentonng that f we replace βσ by Σ n Eq ] 6), whch s permssable n nonrelatvstc cases, we obtan [Σ p,h 0 = 0 Thus, n the nonrelatvstc lmt, D = 0 Ths s a well-known result referred to as Schff s theorem 2 Lowest Approxmaton In the lowest order many-body perturbaton theory, one may show that for an atom wth one electron beyond closed shells, D =2 n>f v v z n n h edm v v ɛ v ɛ n +2 a, n>f n v a z n n h edm a ɛ a ɛ n, where h edm = 2dcβγ 5 p 2 Sums over closed subshells n n the second term vansh Therefore, only those terms wth n n the partally occuped valence subshell contrbute Summng n over the entre valence subshell and subtractng the term wth n = v leads to D =2 n>f v v z n n h edm v v ɛ v ɛ n 2 a a z v v v v h edm a ɛ a ɛ n Snce z and h edm are Hermetan operators, we may re-express the above equaton n the form: D =2 v v z h edm v v, 8) ɛ v ɛ where ranges over all possble one-electron states, both core states a and vrtual ststes n 21 Angular Decomposton We may wrte h edm φ v r) = 2cdp = 2cd 1 d 2 dr 2 r d 2 dr 2 ) ) 1 Gv r) Ω κvm v r F v r) Ω κvm v κvκv1) r 2 ) F v r) Ω κvm v κvκv+1) r 2 ) G v r) Ω κvm v 4

5 Table 1: Drac-Hartree-Fock and RPA calculatons of sheldng factors D/d for alkal-metal atoms and Au Atom State Z D DHF /d DW RPA /d DRPA L 2s 1/ Na 3s 1/ K 4s 1/ Rb 5s 1/ Cs 6s 1/ Au 6s 1/ W +Z /d The matrx element of h edm s therefore wth nm n h edm v v =2cdδ κn κ v δ mn v n h eff v, 9) n h eff v =2cd [ d 2 F v dr G n 0 dr 2 κ vκ v 1) d 2 G v r 2 G n F v + F n dr 2 κ ] vκ v +1) r 2 F n G v 10) Smlarly, wth v v v z n v = v z n 11) 2 v + 1) v +1) v z n = κ v C 1 κ v 0 rdr [G v G n + F v F n ] 12) The expresson for the atomc dpole moment s therefore v v z h eff v D =2 13) 2 v + 1) v +1) ɛ v ɛ In Table 1, we present DHF values of the dpole enhancement factor for alkal-metal atoms The forth column contans values DW RPA /d wth RPA correctons to matrx elements of h edm and the ffth column contans values DW RPA +Z /d that nclude RPA correctons to matrx elements of both h edm and z 5